Adaptive Iterative Soft-Thresholding Algorithm with the Median Absolute Deviation

TL;DR

This paper provides a theoretical analysis of the adaptive ISTA algorithm using median absolute deviation for noise estimation, including convergence properties and stability analysis.

Contribution

It offers the first theoretical insights into adaptive ISTA with median absolute deviation, detailing fixed point properties and convergence guarantees.

Findings

Proves local linear convergence of the algorithm.

Analyzes fixed point properties such as scale equivariance.

Demonstrates global convergence behavior.

Abstract

The adaptive Iterative Soft-Thresholding Algorithm (ISTA) has been a popular algorithm for finding a desirable solution to the LASSO problem without explicitly tuning the regularization parameter $\lambda$. Despite that the adaptive ISTA is a successful practical algorithm, few theoretical results exist. In this paper, we present the theoretical analysis on the adaptive ISTA with the thresholding strategy of estimating noise level by median absolute deviation. We show properties of the fixed points of the algorithm, including scale equivariance, non-uniqueness, and local stability, prove the local linear convergence guarantee, and show its global convergence behavior.